98 BELL SYSTEM TECHNICAL JOURNAL 



F{jx). With this determined the required impedance Zg is simply 

 by (4) 



Z, = ZJ^ + E,/ = (10) 



on the assumption that a' = a-N^a is quite small compared to unity. 

 This gives formula (1). 



Note: The expansion (7), which is believed to be novel, was derived 

 by a limiting process rather too long and unsuitable for inclusion in 

 this paper. It and the following additional expansions are quite 

 useful in certain problems on wave propagation. 



COS d-Ki{pi-\i) = — -vl I ^-."'V^^+i QQg ^^'jj^ d^^ 



Jo 



e-Ki{p'i-y[i) = - \1 f ^ — . c-"'^'^^^ s'mx'fxdp, 

 Jo a'm- + t 



sm 

 where 



COS 6 = y/p = j/V.v^ + A'-, 



sin 6 = x/p = .x'/V-v^ + 3'". 



